Final answer:
To show that P(y)=(λʸye−λ)/y! y=0,1,2,…,λ>0 is a valid probability distribution function, we need to check two things: 1) the function is non-negative for all possible values of y, and 2) the sum of the probabilities for all possible values of y is equal to 1.
Step-by-step explanation:
A random variable Y is said to have a Poisson probability distribution if and only if P(y)=(λʸye−λ)/y! y=0,1,2,…,λ>0. To show that this is a valid probability distribution function, we need to check two things: 1) the function is non-negative for all possible values of y, and 2) the sum of the probabilities for all possible values of y is equal to 1.
1) To show that the function is non-negative, we can consider the factors separately. λʸ is always non-negative since λ>0 and y is a non-negative integer. e−λ is also non-negative for any value of λ. Finally, y! is non-negative since it is the product of non-negative integers. Therefore, all factors are non-negative, and the function is non-negative as well.
2) To show that the sum of the probabilities is equal to 1, we can consider the infinite sum of the function. However, since the sum of the probabilities for all possible values of y is infinite, we cannot directly compute the sum. Instead, we can use a mathematical property of the Poisson distribution which states that the sum of the probabilities is equal to 1. This property can be proved using calculus and the Taylor series expansion of the exponential function. Therefore, we can conclude that the function is a valid probability distribution.